then the sum of the successive DTFTs over time equals the DTFT of the
whole signal
:

(9.3)

Consequently, the inverse-STFT is simply the inverse-DTFT of this sum:

We may now introduce spectral modifications by multiplying each
spectral frame
by some filter frequency response
to get

(9.4)

Note that
can be different for each frame, giving a certain
class of time-varying filters. The filtered output signal spectrum
is then

(9.5)

so that

(9.6)

where

(9.7)

This chapter discusses practical implementation of the above
relations using a Fast Fourier Transform (FFT). In particular, we
use an FFT to compute efficiently what may be regarded as a
sampled DTFT. We will look at how sampling density
must be increased along the unit circle when spectral modifications
are to be performed, and we will discuss further the COLA condition on
the analysis window and hop-size.
In the end, our practical FFT-convolution engine will look as follows:

(9.8)

The sum over
may be interpreted as adding separately filtered
frames
. Due to this filtering, the frames must
overlap, even when the rectangular window is used. As a result, the
overall system is often called an
overlap-add FFT processor,
or
``OLA processor'' for short. It is regarded as a sequence of FFTs
which may be modified, inverse-transformed, and summed. This
``hopping transform'' view of the STFT is the Fourier dual of the
``filter-bank'' interpretation to be discussed in Chapter 9.